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Which is theIs there a name for an associative algebra, which is further endowed with a Lie algebra structure such that the Leibniz rule holds, i.e., $$[a, b \circ c]=[a,b]\circ c+b\circ [a,c],$$$$[a, b \circ c]=[a,b]\circ c+b\circ [a,c], \hspace{15mm}(*)$$ where $[,]$ is the Lie algebra commutator (Lie bracket) and $\circ$ is the multiplication in our algebra?

If the multiplication $\circ$ is commutative, then such an algebra is called a Poisson algebra but I was not able to find in the literature the name for the case when the commutativity assumption is dropped, although the notion itself seems fairly natural.

Indeed, any associative algebra is readily endowed with a Lie algebra structure with the commutator $[a,b]=a\circ b-b\circ a$ which obeys the Leibniz rule $(*)$ but there could exist other Lie algebra structures which still obey $(*)$.

Any relevant references are greatly appreciated.

  Thanks in advance.

Which is the name for an associative algebra, which is further endowed with a Lie algebra structure such that the Leibniz rule holds, i.e., $$[a, b \circ c]=[a,b]\circ c+b\circ [a,c],$$ where $[,]$ is the Lie algebra commutator (Lie bracket) and $\circ$ is the multiplication in our algebra?

If the multiplication $\circ$ is commutative, then such an algebra is called a Poisson algebra but I was not able to find in the literature the name for the case when the commutativity assumption is dropped. Any relevant references are greatly appreciated.

  Thanks in advance.

Is there a name for an associative algebra which is further endowed with a Lie algebra structure such that the Leibniz rule holds, i.e., $$[a, b \circ c]=[a,b]\circ c+b\circ [a,c], \hspace{15mm}(*)$$ where $[,]$ is the Lie algebra commutator (Lie bracket) and $\circ$ is the multiplication in our algebra?

If the multiplication $\circ$ is commutative, then such an algebra is called a Poisson algebra but I was not able to find in the literature the name for the case when the commutativity assumption is dropped, although the notion itself seems fairly natural.

Indeed, any associative algebra is readily endowed with a Lie algebra structure with the commutator $[a,b]=a\circ b-b\circ a$ which obeys the Leibniz rule $(*)$ but there could exist other Lie algebra structures which still obey $(*)$.

Any relevant references are greatly appreciated. Thanks in advance.

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another-guest
another-guest
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Which is the name for an associative algebra, which is further endowed with a Lie algebra structure such that the Leibniz rule holds, i.e., $$[a, b \circ c]=[a,b]\circ c+b\circ [a,c],$$ where $[,]$ is the Lie algebra commutator (Lie bracket) and $\circ$ is the multiplication in our algebra?

Note that ifIf the multiplication $\circ$ is commutative, then such an algebra is called a Poisson algebra but I was not able to find in the literature the name for the case when the commutativity assumption is dropped. Any relevant references are greatly appreciated.

Thanks in advance.

Which is the name for an associative algebra, which is further endowed with a Lie algebra structure such that the Leibniz rule holds, i.e., $$[a, b \circ c]=[a,b]\circ c+b\circ [a,c],$$ where $[,]$ is the Lie algebra commutator (Lie bracket) and $\circ$ is the multiplication in our algebra?

Note that if the multiplication $\circ$ is commutative, then such an algebra is called a Poisson algebra.

Which is the name for an associative algebra, which is further endowed with a Lie algebra structure such that the Leibniz rule holds, i.e., $$[a, b \circ c]=[a,b]\circ c+b\circ [a,c],$$ where $[,]$ is the Lie algebra commutator (Lie bracket) and $\circ$ is the multiplication in our algebra?

If the multiplication $\circ$ is commutative, then such an algebra is called a Poisson algebra but I was not able to find in the literature the name for the case when the commutativity assumption is dropped. Any relevant references are greatly appreciated.

Thanks in advance.

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Let $L$ beWhich is the name for an associative algebra over a field $\mathbb{F}$, and $L$which is further endowed with a Lie algebra structure such that the Leibniz rule holds, i.e., $$[a, b \circ c]=[a,b]\circ c+b\circ [a,c],$$ where $[,]$ is the Lie algebra commutator (Lie bracket) and $\circ$ is the multiplication in $L$.our algebra?

Is there a name for such an $L$?

Note that if the multiplication $\circ$ is commutative, then $L$such an algebra is called a Poisson algebraPoisson algebra.

Let $L$ be an associative algebra over a field $\mathbb{F}$, and $L$ is further endowed with a Lie algebra structure such that the Leibniz rule holds, i.e., $$[a, b \circ c]=[a,b]\circ c+b\circ [a,c],$$ where $[,]$ is the Lie algebra commutator (Lie bracket) and $\circ$ is the multiplication in $L$.

Is there a name for such an $L$?

Note that if the multiplication $\circ$ is commutative, then $L$ is called a Poisson algebra.

Which is the name for an associative algebra, which is further endowed with a Lie algebra structure such that the Leibniz rule holds, i.e., $$[a, b \circ c]=[a,b]\circ c+b\circ [a,c],$$ where $[,]$ is the Lie algebra commutator (Lie bracket) and $\circ$ is the multiplication in our algebra?

Note that if the multiplication $\circ$ is commutative, then such an algebra is called a Poisson algebra.

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another-guest
another-guest
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